Monte Carlo Integration With Acceptance-Rejection

This article considers Monte Carlo integration under rejection sampling or Metropolis-Hastings sampling. Each algorithm involves accepting or rejecting observations from proposal distributions other than a target distribution. While taking a likelihood approach, we basically treat the sampling scheme as a random design, and define a stratified estimator of the baseline measure. We establish that the likelihood estimator has no greater asymptotic variance than the crude Monte Carlo estimator under rejection sampling or independence Metropolis-Hastings sampling. We employ a subsampling technique to reduce the computational cost, and illustrate with three examples the computational effectiveness of the likelihood method under general Metropolis-Hastings sampling.